1211.2676 (Davide Masoero et al.)

A deformation of the method of characteristics and the Cauchy problem
for Hamiltonian PDEs in the small dispersion limit    [PDF]

Davide Masoero, Andrea Raimondo

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the 'variational string equation', a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian PDE satisfies the appropriate string equation in the small dispersion limit. Finally, we apply our construction to compute explicitly the first two perturbative corrections of the solution of the general Hamiltonian PDE.

View original: http://arxiv.org/abs/1211.2676